For fried leeks, heat 1 inch oil to 350 degrees F in a heavy pot. Meanwhile, in a bowl, stir flour with 1/4 teaspoon each salt and pepper. In another bowl, pour buttermilk. Working in about 4 batches, coat leek rounds in buttermilk, then dredge in flour. Shake off excess flour and fry leeks until golden, about 1 minute. Transfer to paper towels. Repeat with remaining leek rounds.

For butter sauce, place white wine, vinegar, celery, chopped leeks, remaining 1/4 teaspoon each salt and pepper, and sugar in a pot over medium-high heat. Bring to a boil and cook until liquid is reduced by two-thirds. Whisk in butter, a few pieces at a time, adding more only when fully incorporated into sauce.

Transfer steamed green beans to platter, drizzle with butter sauce, and top with fried leeks.